When to use PID regulator

Hi,
I have a very basic conceptual question. I have a system (eg. inverted pendulum) that I want to control. So I need to build a controller. That is clear. If I understand it well, I have minimally two separate options:
1 - use PID regulator/controller
2 - analyze the system, linearize it about a specific point so I create LTI system and based on this design my own closed loop controller.
3 - ?

But, when and why should I choose first or second option? What are advantages and disadvantages of each approach? What are other possible solutions?

You've laid out the basic controller design steps in option 2 and you can use PID as the controller of the closed-loop system. Once you have an LTI system to deal with, you can determine from the transfer functions how big to make your PID parameters such that it approximates an ideal system. This is so you can control things like rise time, settling time, gain, etc.

So, I create LTI system in order to determine PID parameters. But I also have an option to create a simple controller based on closed loop formula u = -K*x, right? Or this formula is just a proportional controller, that means it is P letter from PID regulator?
And how is all this related to observer?
Thx

Sure, you can just have a proportional controller which would be like a u=-K*x formula, but the PID usually does better. You can control your system a lot more with PID. I'm not sure how exactly how this relates to observer.